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Properties

Label 2-56e2-1.1-c1-0-31
Degree 22
Conductor 31363136
Sign −1-1
Analytic cond. 25.041025.0410
Root an. cond. 5.004105.00410
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 6·9-s − 11-s + 2·13-s + 3·15-s − 3·17-s − 5·19-s + 3·23-s − 4·25-s − 9·27-s + 6·29-s − 31-s + 3·33-s + 5·37-s − 6·39-s + 10·41-s − 4·43-s − 6·45-s + 47-s + 9·51-s + 9·53-s + 55-s + 15·57-s − 3·59-s + 3·61-s − 2·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s + 0.554·13-s + 0.774·15-s − 0.727·17-s − 1.14·19-s + 0.625·23-s − 4/5·25-s − 1.73·27-s + 1.11·29-s − 0.179·31-s + 0.522·33-s + 0.821·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.894·45-s + 0.145·47-s + 1.26·51-s + 1.23·53-s + 0.134·55-s + 1.98·57-s − 0.390·59-s + 0.384·61-s − 0.248·65-s + ⋯

Functional equation

Λ(s)=(3136s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3136s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31363136    =    26⋅722^{6} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 25.041025.0410
Root analytic conductor: 5.004105.00410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3136, ( :1/2), −1)(2,\ 3136,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
good3 1+pT+pT2 1 + p T + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−T+pT2 1 - T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−3T+pT2 1 - 3 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.230315165473213095839519633705, −7.38315583202405666291782015581, −6.56899809479189996756800142405, −6.11960412337872958718479712327, −5.29913325153166278991319869627, −4.49040947012230363312554161259, −3.93566966697450588624507732655, −2.44876208464128486413690607456, −1.08713788900686923431368242918, 0, 1.08713788900686923431368242918, 2.44876208464128486413690607456, 3.93566966697450588624507732655, 4.49040947012230363312554161259, 5.29913325153166278991319869627, 6.11960412337872958718479712327, 6.56899809479189996756800142405, 7.38315583202405666291782015581, 8.230315165473213095839519633705

Graph of the ZZ-function along the critical line